Entire solutions of the Allen-Cahn equation
Abstract：The Allen-Cahn equation arises in the gradient theory of phase transitions. In this lecture, we will discuss the existence and classification of the entire solutions of the Allen-Cahn type equations. As we will see, this type of equations are closely related to the theory of minimal surfaces and integrable systems, such as Toda system and elliptic sine-Gordon equation. It turns out that the structure of the solution space of the Allen-Cahn type equation is now better understood in dimension two than higher dimensions. For instance, the finite Morse index solutions of the elliptic sine-Gordon equation can be completed classified in dimension two. We will also mention some questions which remain to be answered in this area.